Homework 1 - Truth Tables & Linear Programming with AMPL

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1 Problem 1

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2 Problem 2

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3 Problem 3

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3.1 Task a

3.1.1 Decision Variables

bondA: dollars \(\in \mathbb{R}\) to invest in bond A
bondB: dollars \(\in \mathbb{R}\) to invest in bond B
bondC: dollars \(\in \mathbb{R}\) to invest in bond C
bondD: dollars \(\in \mathbb{R}\) to invest in bond D
bondE: dollars \(\in \mathbb{R}\) to invest in bond E

3.1.2 Objective Function

• Maximize the Expected Earnings of the portfolio

\[ Maximize \ Z = (0.043 \times bondA) + (0.027 \times bondB) + (0.025 \times bondC) + (0.022 \times bondD) + (0.045 \times bondE) \]

3.1.3 Constraints

C1: Budget to invest is $10 MM or less \[ budget: bondA + bondB + bondC + bondD + bondE \leq 10 \]

C2: At least $4 million must be invested in government and agency bonds \[ govtAndAgency: bondB + bondC + bondD \geq 4 \]

C3: Average Quality of the Portfolio must not exceed 1.4 \[ avgQuality: (0.6 \times bondA) + (0.6 \times bondB) - (0.4 \times bondC) - (0.4 \times bondD) + (3.6 \times bondE) \leq 0 \]

C4: The Average Maturity must not Exceed Five Years \[ avgMaturity: (4 \times bondA) + (10 \times bondB) - (1 \times bondC) - (2 \times bondD) - (3 \times bondE) \leq 0 \]

3.1.4 Code

3.1.5 Output

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4 Problem 4

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4.1 Task a

4.1.1 Decision Variables

tv = the number of minutes \(\in \mathbb{R}\) to air advertising on the television medium
magazine = the number of pages \(\in \mathbb{I}\) to to advertise on the magazine medium

4.1.2 Objective Function

• Maximize the total audience reach

\[ Maximize \ Z = (1.8\times tv) + (1.0 \times magazine) \]

4.1.3 Constraints

C1: Must not Exceed Budget of 1 Million dollars
\[ budget: (20,000 \times tv) + (10,000 \times magazine) \leq 1,000,000 \]

C2: Must have at least 10 minutes of air time on the TV medium
\[ minTimeTV: tv \geq 10 \]

4.1.4 Code

4.1.5 Output

4.1.6 Solving Problem 4(a) Graphically by Hand

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4.2 Task b

4.2.1 Additional Constraint: Labor Time

C3: Only 100 person weeks available, given it takes three weeks and one week to create a tv and magazine minute for advertisement, respectively.
\[ personWeeks: (3 \times tv) + (1 \times magazine) \leq 100 \]

4.2.2 Code

4.2.3 Output

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4.3 Task c

4.3.1 Additional Constraint: Radio Advertising Medium

4.3.2 Decision Variables

tv = the number of minutes \(\in \mathbb{R}\) to air advertising on the television medium
magazine = the number of pages \(\in \mathbb{I}\) to to advertise on the magazine medium radio = the number of minutes \(\in \mathbb{R}\) to air advertising on the radio medium

4.3.3 Objective Function

• Maximize the total audience reach

\[ Maximize \ Z = (1.80\times tv) + (1.00 \times magazine) + (0.25 \times radio) \]

4.3.4 New Constraints

C1: Must not Exceed Budget of 1 Million dollars \[ budget: (20,000 \times tv) + (10,000 \times magazine) + (2,000 \times radio) \leq 1,000,000 \]

C2: Must have at least 10 minutes of air time on the TV medium
\[ minTimeTV: tv \geq 10 \]

C3: Only 100 person weeks available, given it takes three weeks and one week to create a tv and magazine minute for advertisement, respectively. It only takes one day for radio.
\[ personWeeks: (3 \times tv) + (1 \times magazine) + (\frac{1}{7} \times radio) \leq 100 \]

4.3.5 Code

4.3.6 Output

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4.4 Task d

4.4.1 Additional Constraints: Miminum Magazine and Maximum Radio Requirements

C4: Must sign up for at least 2 magazine pages \[ minMagazines: magazine \geq 2 \]

C5: Must to exceed 120 minutes of radio \[ maxRadio: radio \leq 120 \]

4.4.2 Code

4.4.3 Output

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5 Problem 5

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5.1 Base Mathematical Formulation and Code

• Each task shows a separate change to the base model. Therefore, each change should not accumulate.

5.1.1 Mathematical Formulation

5.1.2 Code for Model .mod and Input Data .dat

5.2 Task a

5.2.1 Changed Constraint for Total Hours

• Change the constraints so that total hours used by all products must equal the total hours available for each stage

5.2.2 Code

5.2.3 Output

There is no difference in the optimal solution because the range of Time before there is a change in optimal remains the same, and the hours available have not changed.

5.3 Task b

5.3.1 New Constraint for Max Weight

• Restrict the total weight of all products to be less than a new parameter, max_weight = 6,500

\[ totalWeight: \sum_{p \ \in \ PROD} Make_{p} \leq max\_weight \]

5.3.2 Code

5.3.3 Output

The total number of tons has reduced from 7,000 to 6,500 per week

5.4 Task c

5.4.1 Changed Objective Function

• Change the objective function to maximize total tons

\[ maximize \ Total\_Tons = \sum_{p \ \in \ PROD} Make_{p} \]

5.4.2 Code

5.4.3 Output

The data file does not make a diference in the optimal (assuming that is what the question is asking). Please note that the total number of tons produced are the same as in the base model; however, the allocation of tons have shifted among each of the products.

5.5 Task d

5.5.1 New Constraint

5.5.2 Code

5.5.3 Output

5.6 Task e

5.6.1 Changing Input Data via .dat File

5.6.2 Code

5.6.3 Output

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6 Problem 6

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6.1 Task a - c

6.1.1 Decision Variables

bondA: dollars \(\in \mathbb{R}\) to invest in bond A
bondB: dollars \(\in \mathbb{R}\) to invest in bond B
bondC: dollars \(\in \mathbb{R}\) to invest in bond C
bondD: dollars \(\in \mathbb{R}\) to invest in bond D
bondE: dollars \(\in \mathbb{R}\) to invest in bond E

6.1.2 Objective Function

• Maximize the Expected Earnings of the portfolio

\[ Maximize \ Z = (0.043 \times bondA) + (0.027 \times bondB) + (0.025 \times bondC) + (0.022 \times bondD) + (0.045 \times bondE) \]

6.1.3 Constraints

C1: Budget to invest is $10 MM or less \[ budget: bondA + bondB + bondC + bondD + bondE \leq 10 \]

C2: At least $4 million must be invested in government and agency bonds \[ govtAndAgency: bondB + bondC + bondD \geq 4 \]

C3: Average Quality of the Portfolio must not exceed 1.4 \[ avgQuality: (0.6 \times bondA) + (0.6 \times bondB) - (0.4 \times bondC) - (0.4 \times bondD) + (3.6 \times bondE) \leq 0 \]

C4: The Average Maturity must not Exceed Five Years \[ avgMaturity: (4 \times bondA) + (10 \times bondB) - (1 \times bondC) - (2 \times bondD) - (3 \times bondE) \leq 0 \]

C5: Only select Bonds A and D (Don’t select B, C, or E) \[ onlyAandB: bondB + bondC + bondE = 0; \]

C6: Municipal Bonds must be less than or equal to $3 MM \[ municipal: bondA \leq 3; \]

6.1.4 Code

6.1.5 Output

6.2 Task d:

You may not borrow more than 2.83%, since that is the expected
yield to maturity (30% of bondA * 4.3%) + (70% of bondD * 2.2%)

6.3 Task e:

If you borrowed at a rate greater than the expected YTM, then the
venture would not be profitable.